An Upper Bound of Singleton Type for Componentwise Products of Linear Codes

We give an upper bound that relates the dimensions of some given number of linear codes, with the minimum distance of their componentwise product. A typical result is as follows: given t linear codes C_i of parameters [n,k_i]_q with full support, one can find codewords c_i ∈ C_i such that 1 ≤ w(c₁*⋯*c_t) ≤ max(t-1, n+t-(k₁+⋯+k_t)).